This course is an important part of the undergraduate stage in education for future economists. It's also useful for graduate students who would like to gain knowledge and skills in an important part of math. It gives students skills for implementation of the mathematical knowledge and expertise to the problems of economics. Its prerequisites are both the knowledge of the single variable calculus and the foundations of linear algebra including operations on matrices and the general theory of systems of simultaneous equations. Some knowledge of vector spaces would be beneficial for a student.
The course covers several variable calculus, both constrained and unconstrained optimization. The course is aimed at teaching students to master comparative statics problems, optimization problems using the acquired mathematical tools.
Home assignments will be provided on a weekly basis.
The objective of the course is to acquire the students’ knowledge in the field of mathematics and to make them ready to analyze simulated as well as real economic situations.
Students learn how to use and apply mathematics by working with concrete examples and exercises. Moreover this course is aimed at showing what constitutes a solid proof. The ability to present proofs can be trained and improved and in that respect the course is helpful. It will be shown that math is not reduced just to “cookbook recipes”. On the contrary the deep knowledge of math concepts helps to understand real life situations.
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Impartido por:

Kirill Bukin

Transcripción

I would like to add some remark, given a critical point. That means that the first derivative turn zero at this point. For example, the second order derivative is negative, we can say more. Then x star is a strict local max. That means that in some neighborhood or an interval, if we withdraw this point then on this interval, the following inequality holds it's strict. Analogously, if at the critical point the second order derivative is positive, this point is a strict local minimum. Now, we proceed with the descriptions of local and global extrema in case or the function of many variables. Let's suppose we have a function, this time this function depends on n variables and let it be defined on some set U, open set in R^n. For our purposes, it's quite sufficient to consider a function of two variables. We start with drawings with the graphs of these functions. So, let's suppose there is a function x squared plus y squared, where the point (x, y) belongs to R two space. How will the graph of this function looks like? This time we have to draw a graph in 3D, that's how we get three X's. So, it's easy to draw the graph, it looks like a bowl. We can say this is a bowl-shaped graph. Looks like that. Well clearly, the origin is the minimum point of this function. Actually, it's a global minimum because if (x, y) is not taken at the origin, then x squared plus y squared is strictly greater than zero, so actually, we can even make the same remark that the origin for this function is the point of the global strict minimum. Now, if we reverse the signs in the definition or the function Z, and we'll make it minus x squared minus y squared, we have to reflect this bowl upside down and the graph this time looks like that. It looks like that, and the origin this time becomes the point of the strict global maximum. What if we take plus since in front of x squared term, and subtract y squared. It's interesting that the surface which is a graph of this function is not that easy to draw compared to this bowl-shaped surfaces. So, in order to draw the graph for this function, let's cut this surface by the coordinate plane. Let's consider that y equals zero, then this term turns zero, and all we have to do, we have to draw the parabola which faces upwards, but if we start cutting this surface with the vertical planes which passes through any point on x axis, then we get a collection of parabolas which face down and that's how the surface looks like. It looks like a saddle. So, you can sit there at the origin and stretch your legs down. That's why the point which is the origin is called a saddle point. Now, it's time to proceed with the formal definitions of maximum, minimum. So, let's suppose our function is defined as earlier on U, which is an open set in Rn, and we'll say, so that will be definition. Now we're defining global maximum. The point x star, well, clearly this is a vector, but I won't indicate this because it will be self evident. Is the global max, if for all x from U, the following inequality holds. What about the local maximum? A point can be considered as a local maximum of such a function if there exists. So, we have finished with that, and now we proceed with the local maximum. So, what we need to add to the previous definition. If there exists a ball above the point B(x*), such that f of x is no greater than f of x, star for all x we intersect with the U set, then we call such a point x star is a local max.